Levy Processes and Stochastic Calculus: A Graduate-Level Textbook on Lévy Processes and Stochastic Calculus by David App

Introduction

For students and researchers navigating the intricate worlds of probability theory and stochastic processes, Levy Processes and Stochastic Calculus by David Applebaum stands as an authoritative and comprehensive guide. Published by Cambridge University Press, this hardbound edition is an essential resource for Indian academics, mathematicians, and quantitative finance professionals who seek a rigorous yet accessible treatment of Lévy processes and their stochastic calculus. The book bridges foundational theory with advanced applications, making it a cornerstone text for graduate-level study and research in India’s growing mathematical sciences community.

Book Overview

This fully revised edition offers a deep exploration of Lévy processes—a rich class of random processes with jumps that generalise Brownian motion and Poisson processes. David Applebaum masterfully connects the theory of Lévy processes with stochastic calculus, providing readers with the tools needed to model systems influenced by random noise, from financial markets to physical phenomena. The text progresses from basic definitions and properties to sophisticated topics like Malliavin calculus and stability theory, ensuring a logical and thorough learning path. With over 500 pages of meticulously crafted content, this book is both a textbook and a reference work for anyone serious about stochastic analysis.

Key Highlights

• Comprehensive Coverage: From fundamental concepts to advanced topics, including regular variation, subexponential distributions, and multiple Wiener-Lévy integrals.
• Rigorous Yet Accessible: Clear explanations of proofs and theorems, including new proofs of Itô representation and martingale representation theorems for general Lévy processes.
• Updated Content: This edition introduces necessary and sufficient conditions for finite moments, characterisation of finite variation Lévy processes, and Kunita’s estimates for moments of Lévy-type stochastic integrals.
• Modern Applications: Includes an introduction to Malliavin calculus and chaos decomposition, essential for contemporary research in stochastic analysis.
• Academic Standard: Published by Cambridge University Press, ensuring high editorial and mathematical accuracy.

Inside the Book

The book is structured to guide readers step-by-step through the theory. Early chapters lay the groundwork with the definition and properties of Lévy processes, including infinite divisibility and the Lévy-Khintchine formula. Subsequent chapters delve into stochastic calculus, covering stochastic integration with respect to Lévy processes, the Itô formula, and change of measure. Later sections explore advanced topics such as regular variation, subexponential distributions, and stability theory for Lévy processes. Each chapter is supplemented with exercises and examples, making it suitable for classroom use in Indian universities as well as self-study.

Key Topics

• Lévy processes: definition, properties, and examples
• Infinite divisibility and the Lévy-Khintchine representation
• Stochastic calculus for semimartingales and Lévy processes
• Itô formula and stochastic differential equations with jumps
• Regular variation and subexponential distributions
• Finite variation Lévy processes and moment conditions
• Kunita’s estimates for stochastic integrals
• Itô representation and martingale representation theorems
• Multiple Wiener-Lévy integrals and chaos decomposition
• Introduction to Malliavin calculus
• Stability theory for Lévy processes

Reader Benefits

By studying this book, readers gain a solid command of both the theoretical underpinnings and practical techniques of Lévy processes and stochastic calculus. The detailed proofs and new results equip researchers with the latest tools for original work. Students will find the logical progression from basics to frontiers immensely helpful for coursework and exam preparation. The book also serves as a valuable reference for professionals in quantitative finance, insurance, and physics who need to model jump processes and random noise. Indian readers will appreciate the clear, no-nonsense writing style that makes complex mathematics approachable.

Learning Outcomes

• Understand the fundamental properties and classifications of Lévy processes
• Master the Lévy-Khintchine formula and its implications
• Develop proficiency in stochastic calculus for processes with jumps
• Apply Itô’s formula and stochastic differential equations in jump settings
• Analyse conditions for finite moments and finite variation in Lévy processes
• Utilise advanced topics like Malliavin calculus and chaos decomposition
• Gain familiarity with stability theory and its applications

Who Should Read

This book is ideal for graduate students in mathematics, statistics, and financial engineering at Indian universities. It is also highly recommended for researchers in stochastic processes, probability theory, and mathematical finance. Professionals working in quantitative analysis, risk management, and actuarial science will find the content directly applicable to modelling real-world phenomena with jumps. Additionally, physicists and engineers dealing with random noise and stochastic systems will benefit from the rigorous yet practical treatment.

About the Author

David Applebaum is a renowned mathematician and professor specialising in probability theory and stochastic processes. With decades of teaching and research experience, he has contributed significantly to the theory of Lévy processes and their applications. His clear exposition and deep insights make this book a trusted resource worldwide. Applebaum’s work is widely cited in both pure and applied mathematics, and he is known for making advanced topics accessible to a broad audience.

About the Publisher

Cambridge University Press is one of the oldest and most prestigious academic publishers in the world. Known for its rigorous editorial standards and commitment to scholarly excellence, Cambridge University Press publishes authoritative texts across all fields of science, mathematics, and humanities. This hardcover edition reflects the publisher’s dedication to quality, making it a durable and valuable addition to any academic library in India.

Conclusion

Levy Processes and Stochastic Calculus is an indispensable resource for anyone seeking a deep and modern understanding of stochastic processes with jumps. David Applebaum’s masterful synthesis of theory and advanced topics, combined with the trusted Cambridge University Press imprint, ensures that this book will remain a classic reference for years to come. Whether you are a student, researcher, or professional, this hardbound volume will elevate your knowledge and skills in one of the most dynamic areas of mathematics. Order your copy from Bookshops.in today and embark on a rigorous journey into the world of Lévy processes.